A. Single spell duration data
We have assumed until now that individual heterogeneity across price spells can be fully captured by covariates, either constant or time-varying. It may also be the case that there are other factors affecting the duration of spells in a systematic way but which we do not observe. It may then be worth to account for such unobserved heterogeneity. The most usual way to do it is to assume that this heterogeneity factor has a particular distribution. Consider that the conditional hazard is given by
hi(t | {zis}t0i, θi) = θi htexp [zitα]
where θi is a random variable independent of time-varying covariates zit. The conditional survivor function given observed and unobserved heterogeneity may be derived as
S(t | {zis}t0i, θi) = exp
− t
0hi(s | {zis}t0i, θi) ds
To estimate this model, we have to “integrate out” this conditional survivor function over θi whose density function is µ(θ):
S(t| {zis}t0i) =
∞
0 S(t | {zis}t0i, θi)µ(θi) dθi
Following Lancaster (1979) and Meyer (1990), we can assume that θ is Gamma distributed with mean 1 and variance σ2. Then we get:
S(t| {zis}t0i) = [1 + σ2H0(t| {zis}t0i)]−1/σ2 where:
H0(t| {zis}t0i) = t
0htexp [zisα] ds
Then, if the hazard function is piecewise constant, the log-likelihood can be written as:
ln L = N
i=1
ln
S(ti− 1| {zis}t0i−1) − ciS(ti| {zis}t0i)
=
N i=1
ln
[1 + σ2H0(ti− 1| {zis}t0i−1)]−1/σ2
−ci[1 + σ2H0(ti| {zis}t0i)]−1/σ2
namely:
If the hazard function is constant over time, the likelihood function becomes:
ln L =N
In the general case with K competing risks, if we assume that there exist K unobserved heterogeneity terms θik associated with the K latent durations Tik
and that these K unobserved heterogeneity terms are mutually independent, then the expressions of the K log-likelihood sub-functions Lk are :
ln Lk =N for the model with a piecewise constant hazard function, and:
ln Lk =N
for the model with a constant hazard function. The total log-likelihood function is then:
ln L =3 k=1ln Lk
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Table 1 : Price spell durations
# spells Proportion Proportion Sectoral Average Average spells spells CPI duration duration (weighted) weights(*) (weighted)
All spells 164626 1.000 1.000 1.00 7.44 8.22 By sector
Food 65525 0.398 0.271 0.206 6.31 6.62
Energy 6591 0.04 0.093 0.084 2.60 2.01
Clothes 18216 0.111 0.060 0.092 7.00 7.32
Durable 10371 0.063 0.062 0.060 6.55 6.46
Other manuf. goods 33917 0.206 0.189 0.169 7.49 8.11
Services 30006 0.182 0.326 0.389 11.48 11.86
By outlet type
Hypermarkets 34359 0.209 0.167 0.199 4.97 4.59 Supermarkets 29882 0.182 0.124 0.136 5.74 5.59 Traditional corner shops 39364 0.239 0.254 0.278 9.07 9.12
Services 23883 0.145 0.284 0.143 11.38 11.46
Others 37138 0.226 0.012 0.244 6.81 6.91
By outcome
Price increases 51284 0.312 0.356 - 7.40 8.43 Price decreases (exc. sales) 32046 0.195 0.185 - 4.92 4.99 Price decreases (sales only) 1842 0.011 0.008 - 6.21 6.70 Product replacement 35333 0.215 0.178 - 7.81 8.86 Right-censored spells 44121 0.268 0.273 - 9.06 9.73 Notes: Average duration is in months.
The coverage of "service" outlet type and of the "service" sector are distinct.
The column weight report the CPI weight of components for sectors.
In the case of outlets, it reports the proportion of price quotes by type of outlet
Source: individual price records used for the calculation of the French CPI (INSEE, 1994-2003)
Table 2 : Descriptive statistics for estimated models
All Price Price Product
outcomes increases decreases replacements
# models 734 309 228 197
# spells used per model 362.18 350.85 360.86 381.49
By type of good
Food 343 158 131 54
Energy 26 14 12 0
Durable goods 57 15 18 24
Clothes 55 8 0 47
Other manuf. goods 169 64 51 54
Services 84 50 16 18
By type of outlet
Hypermarkets 204 79 79 46
Supermarkets 183 76 70 37
Traditional corner shops 149 61 27 61
Services 73 41 17 15
Others 125 52 35 38
Source: individual price records used for the calculation of the French CPI (INSEE, 1994-2003)
Table 3. Estimated parameters of a competing-risks piecewise constant hazard model Item: Haircut for women
Hazard for price increases Hazard for price decreases
Parameters Estimates St. errors Estimates St. errors
Baseline hazard (log)
b1 -5.1399 0.463 -4.9263 0.4853
b2 -4.7192 0.3855 -5.631 0.6431
b3 -4.1872 0.336 - .
b4 -4.0376 0.3216 -5.3919 0.7383
b5 -3.6239 0.2826 -4.4517 0.5417
b6 -4.0041 0.3395 - .
b7 -3.6818 0.3228 -4.8021 0.7441
b8 -3.7113 0.3396 - .
b9 -3.998 0.3832 -5.4186 1.0469
b10 -3.6379 0.3416 -5.1609 1.0439
b11 -3.8749 0.3879 -4.4184 0.7838
b12 -2.988 0.2723 -4.8801 1.0592
b13 -3.17 0.3261 -3.3345 0.6313
b14 -3.3877 0.1856 -4.533 0.7054
Time-varying covariates
Cumulated inflation 3.7282 3.7607 -35.5123 20.5645
January 2002 (Euro changeover) 3.1075 0.265 4.0545 0.6251
August 1995 (VAT) 1.6127 0.4644 1.3325 1.0354
September 1995 (VAT) 1.8886 0.3615 1.1493 1.0698
April 2000 (VAT) 0.1701 0.6456 1.0739 1.0272
Wald test for constant hazard
Wald stat. p-value Wald stat. p-value
b1 =...= b12 31.93 0.002 10.313 0.413
b2 =…= b12 17.88 0.057 3.285 0.857
b1 =…= b5 = b7 =...= b12 26.353 0.003 3.314 0.913
b1 =…= b11 14.618 0.147 3.313 0.855
b2 =…= b11 7.805 0.554 3.285 0.772
Number of spells: 563 563
Log-likelihood at the maximum: -946.79 -191.75
Notes:
Coicop code (6 digit) is 121112
le 4. Wald tests for the assumption of a constant hazard: % of non rejection across strata Price Price Price Price Product Product increasesincreasesdecreasesdecreasesreplacementsreplacements weightedweightedweighted verall non-rejection at 5%0.6250.3450.7850.6140.7210.688 non-rejection at 1%0.7060.4430.8420.6750.7510.731 non-rejection at 10%0.5630.3070.7020.5540.690.669 non-rejection: h2=…=h110.8610.6900.9470.8760.7410.748 y type of good 0.6840.5160.7630.6570.9630.957. nergy0.4290.1890.50.206-- able goods0.8000.7420.8890.8090.8750.826 othes0.8750.858--0.1060.059 ther manuf. goods0.7190.4740.9020.7160.9070.882 ervices0.2800.1040.6880.6470.8330.589 y type of outlet permarkets0.7590.5960.7720.6470.7610.744 upermarkets0.6840.5470.8290.7220.9730.970 tional corner shops0.4590.2200.5930.4020.5080.551 ervices0.2440.1050.6470.5860.8670.622 thers0.8270.6870.9430.8100.7110.818 te: es are percentages of cases for which a Wald test at the 5% level does not reject H0: h1=…=h14 ource: individual price records used for the calculation of the French CPI (INSEE, 1994-2003)
Table 5: Shape of the hazard in strata in which the assumption of a constant hazard is rejected Panel A: by sector
# strata % of increasing % of increasing % with peaks % with peaks hazards hazards at 1, 6 or 12 at 1, 6 or 12
weighted weighted
Price Increases
All sectors 116 0.491 0.634 0.647 0.533
Food 50 0.160 0.198 0.660 0.567
Energy 8 0.125 0.049 0.625 0.784
Durable goods 3 0.667 0.266 1.000 1.000
Clothes 1 1.000 1.000 0.000 0.000
Other manuf. Goods 18 0.778 0.851 0.667 0.428
Services 36 0.861 0.922 0.611 0.468
Price Decreases
All sectors 49 0.082 0.178 0.837 0.718
Food 31 0.000 0.000 0.935 0.900
Energy 6 0.000 0.000 1.000 1.000
Durable goods 2 0.000 0.000 1.000 1.000
Clothes - - - -
-Other manuf. Goods 5 0.400 0.652 0.200 0.034
Services 5 0.400 0.408 0.600 0.475
Product Replacements
All sectors 55 0.836 0.486 0.145 0.224
Food 2 0.000 0.000 0.500 0.561
Energy - - - -
-Durable goods 3 0.667 0.220 0.667 0.932
Clothes 42 0.929 0.898 0.000 0.000
Other manuf. goods 5 0.800 0.663 0.600 0.621
Services 3 0.333 0.095 0.667 0.238
Source: individual price records used for the calculation of the French CPI (INSEE, 1994-2003)
Table 5: Shape of the hazard in strata in which the assumption of a constant hazard is rejected Panel B: by outlet type
# strata % of increasing % of increasing % with peaks % with peaks hazards hazards at 1, 6 or 12 at 1, 6 or 12
weighted weighted
Price Increases
All sectors 116 0.491 0.634 0.647 0.533
Hypermarkets 19 0.000 0.000 0.684 0.805
Supermarkets 24 0.170 0.112 0.625 0.583
Traditional corner shop 33 0.670 0.600 0.667 0.525
Services 31 0.840 0.925 0.613 0.481
Others 9 0.560 0.793 0.667 0.292
Price Decreases
All sectors 49 0.082 0.178 0.837 0.718
Hypermarkets 18 0.056 0.087 0.833 0.777
Supermarkets 12 0.000 0.000 1.000 1.000
Traditional corner shops 11 0.091 0.248 0.818 0.746
Services 6 0.333 0.354 0.667 0.544
Others 2 0.000 0.000 -
-Product Replacements
All sectors 55 0.836 0.486 0.145 0.224
Hypermarkets 11 0.818 0.647 0.091 0.252
Supermarkets 1 1.000 1.000 1.000 1.000
Traditional corner shops 30 0.800 0.639 0.167 0.314
Services 2 0.500 0.111 0.500 0.111
Others 11 1.000 1.000 0.000 0.000
Source: individual price records used for the calculation of the French CPI (INSEE, 1994-2003)
Table 6: Student test results on estimated parameters associated with accumulated inflation % positive % positive % negative % negative % positive % positive and and and and and and significantsignificantsignificantsignificantsignificantsignificant parameterparameterparameterparameterparameterparameter weightedweightedweighted All sectors0.4530.4350.1220.1980.1370.214 By type of good Food0.6010.6020.0990.1170.1110.119 Energy0.5710.7580.1670.358-- Durable goods0.2000.2590.0560.1020.2080.248 Clothes0.0000.000--0.1700.176 Other manuf. goods0.2970.2070.1540.1310.0930.063 Services0.3000.3710.2500.3520.1670.404 By type of outlet Hypermarkets0.4560.4810.1000.1160.0430.025 Supermarkets0.5660.5690.1000.1030.1350.137 Traditional corner shops0.4260.5000.1850.2930.2300.266 Services0.2930.3480.2350.3590.0670.336 Others0.4420.2590.1140.1150.1320.110 Note: Models are estimated with a piecewise constant hazard function Source: individual price records used for the calculation of the French CPI (INSEE, 1994-2003)
Price IncreasesPrice DecreasesProduct Replacements
Table 7: Average value of estimated parameters associated with the euro cash changeover dummy % positive % positive % positive % positive % positive % positive and and and and and and significantsignificantsignificantsignificantsignificantsignificant parameterparameterparameterparameterparameterparameter weightedweightedweighted All sectors0.4470.5850.3740.4540.2750.439 By type of good Food0.2600.2840.3180.3100.1670.122 Energy0.2220.1750.1000.1340.2860.331 Durable goods0.5710.6410.5000.5620.1070.088 Clothes0.8330.838--- Other manuf. goods0.3750.5070.3780.5720.4120.495 Services0.8570.8770.7500.7450.4440.677 By type of outlet Hypermarkets0.1040.0780.1630.1390.3600.383 Supermarkets0.1600.1220.3190.2710.1250.054 Traditional corner shops0.7970.7820.6670.6270.2380.418 Services0.7890.8120.6670.6940.4670.665 Others0.3550.5360.4000.5060.2800.221 Note: Models are estimated with a piecewise constant hazard function Source: individual price records used for the calculation of the French CPI (INSEE, 1994-2003)
Price IncreasesPrice DecreasesProduct Replacements
Table 8: Average value of estimated parameters associated with the VAT change dummy variables % positive % positive % positive % positive and and and and significantsignificantsignificantsignificant parameterparameterparameterparameter weightedweighted All sectors0.5630.5640.6790.541 By type of good Foodn.c.n.c.n.c.n.c. Energy0.4290.3700.1110.026 Durable goods0.0000.0000.5880.552 Clothes0.5000.489-- Other manuf. goods0.6960.7340.8890.883 Services0.5250.5820.7140.563 By type of outlet Hypermarkets0.5910.4130.7920.555 Supermarkets0.7690.6130.8000.632 Traditional corner shops0.6130.6230.4710.457 Services0.4410.5590.5710.571 Others0.5260.5960.7500.803 Source: individual price records used for the calculation of the French CPI (INSEE, 1994-2003) Note: Models are estimated with a piecewise constant hazard function . Results do not include food items since they are not covered by the main VAT rate. n.c. : not concerned VAT decrease (2000:4) dummy Impact on Price IncreasesPrice Decreases
VAT increase (1995:8) dummy Impact on
dence on duration and state dependence Non state- dependenceState- dependenceNon state- dependenceState- dependenceNon state- dependenceState- dependence models (%) e37.924.669.78.856.915.2 e16.820.718.03.517.810.2 models (%, weighted) e23.211.252.19.354.414.3 e33.332.228.110.512.918.4 ce records used for the calculation of the French CPI (INSEE, 1994-2003) each destination reports the cross-tabulation of models according to two tests results. The first row of each panel ("non-duration the breakdown of models for which the null of non-duration dependence (i.e. h1=...=h14) is not rejected at the 5 percent level. h panel ("non state-dependence") reports the breakdown of models for which the null of non-state dependence (i.e.α1=0, where umulative inflation) is not rejected at the 5 percent level. "N=" reports the number of estimated models.
Price increasesPrice decreasesProduct replacements N=309N=228N=197
European Central Bank working paper series
For a complete list of Working Papers published by the ECB, please visit the ECB’s website (http://www.ecb.int)
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522 “Price setting behaviour in Spain: evidence from micro PPI data” by L. J. Álvarez, P. Burriel and I. Hernando, September 2005.
523 “How frequently do consumer prices change in Austria? Evidence from micro CPI data”
by J. Baumgartner, E. Glatzer, F. Rumler and A. Stiglbauer, September 2005.
524 “Price setting in the euro area: some stylized facts from individual consumer price data”
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535 “The pricing behaviour of firms in the euro area: new survey evidence” by S. Fabiani, M. Druant, I. Hernando, C. Kwapil, B. Landau, C. Loupias, F. Martins, T. Y. Mathä, R. Sabbatini, H. Stahl and A. C. J. Stokman, October 2005.
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